In a large bowl, combine potatoes, shredded cheese, bacon and green onions. Divide into eighteen ¼ c. portions. Shape each portion around a cheese cube to cover completely, forming a ball. Refrigerate, covered, at least 30 minutes.
In a shallow bowl, mix bread crumbs, Parmesan cheese, salt and pepper. Place egg in a separate shallow bowl. Dip potato balls in egg, then in crumb mixture, patting to help coating adhere.
In an electric skillet or a deep-fat fryer, heat oil to 375℉. Fry potato balls, a few at a time, until golden brown, about 2 minutes. Drain on paper towels.